##
**On the classification of metabelian Lie algebras.**
*(English)*
Zbl 0267.17015

According to a theorem of Levi, in characteristic zero a finite dimensional Lie algebra can be written as the direct sum of a semisimple subalgebra and its unique maximal solvable ideal. If the field is algebraically closed, all semisimple Lie algebras and their representations are classified, and A. I. Mal’cev (Mal’tsev) [Izv. Akad. Nauk SSSR, Ser. Mat. 9, 329–356 (1945; Zbl 0061.05303), Engl. transl. in Am. Math. Soc. Transl. (1) 9, 228–262 (1960)] reduced the classification of complex solvable Lie algebras to several invariants plus the classification of nilpotent Lie algebras. The concern of this paper is the latter problem.

For any Lie algebra \(L\) and positive integer \(n\), \(L^n\) is defined recursively by \(L^1 = L\), \(L^{n+1}= [L,L^n]\). A nilpotent Lie algebra of length \(l\) is one such that \(L^l\neq 0= L^{l+1}\). In addition to the length \(l\), another fundamental invariant of nilpotent Lie algebras is the number \(g = \dim(L/L^2)\) – the least number of elements required to generate \(L\). A “free” nilpotent Lie algebra with the invariants \(l\), \(g\) is constructed. Every other nilpotent Lie algebra with the invariants \(l\), \(g\) is a quotient of this free one, and two quotients, are isomorphic if and only if the corresponding defining ideals (relations) in the free algebra are conjugate under the action of its automorphism group. This group is easily described. For metabelian Lie algebras \(L\) \((l=2)\) this method equates the classification of those \(L\) for which \(g = \dim(L/L^2)\) with the determination of orbits of subspaces of \(V\wedge V\) \((V\) is a \(g\)-dimensional vector space) under the action of \(\mathrm{GL}(V)\).

By an orthogonal complementation argument, classifying orbits of \(p\)-dimensional subspaces is shown to be the same as classifying orbits of the \(p\)-codimensional ones. This result is used to set up a duality theory for metabelian Lie algebras \(L\) with the invariant \(g= \dim(L/L^2)\) – an association \(L\to L^0\) which (i) preserves isomorphisms, and which satisfies (ii) \((L^0)^0 \cong L\), (iii) \(\dim(L^2) + \dim((L^0)^2) = (g^2-g)/2\). In a sense this cuts the classification problem in half.

Furthermore, a canonical isomorphism between \(V\wedge V\) and \(\mathrm{Alt}(V^*)\) (the space of alternating forms on \(V^*)\) is exhibited which induces a bijection between the orbit spaces \(V\wedge V/\mathrm{GL}(V)\) and \(\mathrm{Alt}(V^*)/\mathrm{GL}(V^*)\). Thus the classification problem can be viewed as one of obtaining a simultaneous canonical form for a space of alternating forms.

In particular, determining orbits of \(l\)-dimensional subspaces is the same as computing ranks of alternating forms. Due to classical results of Weierstrass and Kronecker on so-called pencils of matrices, a canonical form (complete set of invariants) is obtained for a 2-dimensional space of alternating forms. The metabelian Lie algebras which are free algebras modulo one or two relations (and their duals) are thus completely classified.

These results are applied in §7 to give a complete classification of metabelian Lie algebras of dimension \(\leq 7\), and nearly complete results for dimension 8. By an algebro-geometric dimension argument it is shown that there are infinitely many isomorphism classes for each dimension \(n\geq 9\).

A survey of known results on the classification of nilpotent algebras and their forms is presented along with some new examples.

The paper concludes with a short section on derivations of nilpotent Lie algebras. Concerning a problem of Jacobson, a large class of nilpotent algebras (including the metabelian ones) having injective derivations is exhibited. This yields as corollaries an easy proof of the algebraicity of metabelian Lie algebras, and a result that an \(n\)-dimensional metabelian algebra \(L\) can be faithfully represented in \((n+1)\)-dimensions.

For any Lie algebra \(L\) and positive integer \(n\), \(L^n\) is defined recursively by \(L^1 = L\), \(L^{n+1}= [L,L^n]\). A nilpotent Lie algebra of length \(l\) is one such that \(L^l\neq 0= L^{l+1}\). In addition to the length \(l\), another fundamental invariant of nilpotent Lie algebras is the number \(g = \dim(L/L^2)\) – the least number of elements required to generate \(L\). A “free” nilpotent Lie algebra with the invariants \(l\), \(g\) is constructed. Every other nilpotent Lie algebra with the invariants \(l\), \(g\) is a quotient of this free one, and two quotients, are isomorphic if and only if the corresponding defining ideals (relations) in the free algebra are conjugate under the action of its automorphism group. This group is easily described. For metabelian Lie algebras \(L\) \((l=2)\) this method equates the classification of those \(L\) for which \(g = \dim(L/L^2)\) with the determination of orbits of subspaces of \(V\wedge V\) \((V\) is a \(g\)-dimensional vector space) under the action of \(\mathrm{GL}(V)\).

By an orthogonal complementation argument, classifying orbits of \(p\)-dimensional subspaces is shown to be the same as classifying orbits of the \(p\)-codimensional ones. This result is used to set up a duality theory for metabelian Lie algebras \(L\) with the invariant \(g= \dim(L/L^2)\) – an association \(L\to L^0\) which (i) preserves isomorphisms, and which satisfies (ii) \((L^0)^0 \cong L\), (iii) \(\dim(L^2) + \dim((L^0)^2) = (g^2-g)/2\). In a sense this cuts the classification problem in half.

Furthermore, a canonical isomorphism between \(V\wedge V\) and \(\mathrm{Alt}(V^*)\) (the space of alternating forms on \(V^*)\) is exhibited which induces a bijection between the orbit spaces \(V\wedge V/\mathrm{GL}(V)\) and \(\mathrm{Alt}(V^*)/\mathrm{GL}(V^*)\). Thus the classification problem can be viewed as one of obtaining a simultaneous canonical form for a space of alternating forms.

In particular, determining orbits of \(l\)-dimensional subspaces is the same as computing ranks of alternating forms. Due to classical results of Weierstrass and Kronecker on so-called pencils of matrices, a canonical form (complete set of invariants) is obtained for a 2-dimensional space of alternating forms. The metabelian Lie algebras which are free algebras modulo one or two relations (and their duals) are thus completely classified.

These results are applied in §7 to give a complete classification of metabelian Lie algebras of dimension \(\leq 7\), and nearly complete results for dimension 8. By an algebro-geometric dimension argument it is shown that there are infinitely many isomorphism classes for each dimension \(n\geq 9\).

A survey of known results on the classification of nilpotent algebras and their forms is presented along with some new examples.

The paper concludes with a short section on derivations of nilpotent Lie algebras. Concerning a problem of Jacobson, a large class of nilpotent algebras (including the metabelian ones) having injective derivations is exhibited. This yields as corollaries an easy proof of the algebraicity of metabelian Lie algebras, and a result that an \(n\)-dimensional metabelian algebra \(L\) can be faithfully represented in \((n+1)\)-dimensions.

Reviewer: Michael A. Gauger

### MSC:

17B30 | Solvable, nilpotent (super)algebras |

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |

15A21 | Canonical forms, reductions, classification |

15A72 | Vector and tensor algebra, theory of invariants |

15A75 | Exterior algebra, Grassmann algebras |

### Citations:

Zbl 0061.05303
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\textit{M. A. Gauger}, Trans. Am. Math. Soc. 179, 293--329 (1973; Zbl 0267.17015)

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### References:

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